A Client Invests 5million Portfolio And Invests 5 Percent Of It In A Money Market Fund Projected To Earn3 Percent Annually. Estimate The Value Of This Portion Of His Portfolio After Sevenyears
Table 1-6 shows how the initial investment of $4,329.48 can actually generate five $1,000 withdrawals over the next five years.
TABLE 1-6 How an Initial Present Value Funds an Annuity
Amount Available Amount
Time at the Beginning of Ending Amount Before Available After
Period the Time Period Withdrawal Withdrawal Withdrawal
1 $4,329.48 $4,329.48(1.05) = $4,545.95 $1,000 $3,545.95
2 $3,545.95 $3,545.95(1.05) = $3,723.25 $1,000 $2,723.25
3 $2,723.25 $2,723.25(1.05) = $2,859.41 $1,000 $1,859.41
4 $1,859.41 $1,859.41(1.05) = $1,952.38 $1,000 $952.38
To interpret Table 1-6, start with an initial present value of $4,329.48 at t = 0. From t = 0 to t = 1, the initial investment earns 5 percent interest, generating a future value of $4,329.48(1.05) = $4,545.95. We then withdraw $1,000 from our account, leaving $4,545.95 - $1,000 = $3,545.95 (the figure reported in the last column for time period 1). In the next period, we earn one year's worth of interest and then make a $1,000 withdrawal. After the fourth withdrawal, we have $952.38, which earns 5 percent. This amount then grows to $1,000 during the year, just enough for us to make the last withdrawal. Thus the initial present value, when invested at 5 percent for five years, generates the $1,000 five-year ordinary annuity. The present value of the initial investment is exactly equivalent to the annuity.
Now we can look at how future value relates to annuities. In Table 1-5, we reported that the future value of the annuity was $5,525.64. We arrived at this figure by compounding the first $1,000 payment forward four periods, the second $1,000 forward three periods, and so on. We then added the five future amounts at t = 5. The annuity is equivalent to $5,525.64 at t = 5 and $4,329.48 at t = 0. These two dollar measures are thus equivalent. We can verify the equivalence by finding the present value of $5,525.64, which is $5,525.64 X (1.05)"5 = $4,329.48. We found this result above when we showed that a lump sum can generate an annuity.
7.5 The Cash Flow Additivity Principle
To summarize what we have learned so far: A lump sum can be seen as equivalent to an annuity, and an annuity can be seen as equivalent to its future value. Thus present values, future values, and a series of cash flows can all be considered equivalent as long as they are indexed at the same point in time.
The cash flow additivity principle—the idea that amounts of money indexed at the same point in time are additive—is one of the most important concepts in time value of money mathematics. We have already mentioned and used this principle; this section provides a reference example for it.
Consider the two series of cash flows shown on the time line in Figure 1-9. The series are denoted A and B. If we assume that the annual interest rate is 2 percent, we can find the future value of each series of cash flows as follows. Series A's future value is $100(1.02) + $100 = $202. Series B's future value is $200(1.02) + $200 = $404. The future value of (A + B) is $202 + $404 = $606 by the method we have used up to this point. The alternative way to find the future value is to add the cash flows of each series, A and B (call it A + B), and then find the future value of the combined cash flow, as shown in Figure 1-9.
FIGURE 1-9 The Additivity of Two Series of Cash Flows t = 0
$100
$100
$200
$200
$300
$300
The third time line in Figure 1-9 shows the combined series of cash flows. Series A has a cash flow of $100 at t = 1, and Series B has a cash flow of $200 at t = 1. The combined series thus has a cash flow of $300 at t = 1. We can similarly calculate the cash flow of the combined series at t = 2. The future value of the combined series (A + B) is $300(1.02) + $300 = $606—the same result we found when we added the future values of each series.
The additivity and equivalence principles also appear in another common situation. Suppose cash flows are $4 at the end of the first year and $24 (actually separate payments of $4 and $20) at the end of the second year. Rather than finding present values of the first year's $4 and the second year's $24, we can treat this situation as a $4 annuity for two years and a second-year $20 lump sum. If the discount rate were 6 percent, the $4 annuity would have a present value of $7.33 and the $20 lump sum a present value of $17.80, for a total of $25.13.
8 SUMMARY
In this chapter, we have explored a foundation topic in investment mathematics, the time value of money. We have developed and reviewed the following concepts for use in financial applications:
• The interest rate, r, is the required rate of return; r is also called the discount rate or opportunity cost.
• An interest rate can be viewed as the sum of the real risk-free interest rate and a set of premiums that compensate lenders for risk: an inflation premium, a default risk premium, a liquidity premium, and a maturity premium.
• The future value, FV, is the present value, PV, times the future value factor, (1 + rf.
• The interest rate, r, makes current and future currency amounts equivalent based on their time value.
• The stated annual interest rate is a quoted interest rate that does not account for compounding within the year.
• The periodic rate is the quoted interest rate per period; it equals the stated annual interest rate divided by the number of compounding periods per year.
• The effective annual rate is the amount by which a unit of currency will grow in a year with interest on interest included.
• An annuity is a finite set of level sequential cash flows.
• There are two types of annuities, the annuity due and the ordinary annuity. The annuity due has a first cash flow that occurs immediately; the ordinary annuity has a first cash flow that occurs one period from the present (indexed at t - 1).
• On a time line, we can index the present as 0 and then display equally spaced hash marks to represent a number of periods into the future. This representation allows us to index how many periods away each cash flow will be paid.
• Annuities may be handled in a similar fashion as single payments if we use annuity factors instead of single-payment factors.
• The present value, PV, is the future value, FV, times the present value factor, (1 + ryN.
• The present value of a perpetuity is A/r, where A is the periodic payment to be received forever.
• It is possible to calculate an unknown variable, given the other relevant variables in time value of money problems.
• The cash flow additivity principle can be used to solve problems with uneven cash flows by combining single payments and annuities.
PROBLEMS 1. The table below gives current information on the interest rates for two two-year and two eight-year maturity investments. The table also gives the maturity, liquidity, and default risk characteristics of a new investment possibility (Investment 3). All investments promise only a single payment (a payment at maturity). Assume that premiums relating to inflation, liquidity, and default risk are constant across all time horizons.
|
Investment |
Maturity (in years) |
Liquidity |
Default Risk |
Interest Rate (%) |
|
l |
2 |
High |
Low |
2.0 |
|
2 |
2 |
Low |
Low |
2.5 |
|
3 |
7 |
Low |
Low | |
|
4 |
8 |
High |
Low |
4.0 |
|
5 |
8 |
Low |
High |
6.5 |
Based on the information in the above table, address the following:
A. Explain the difference between the interest rates on Investment 1 and Investment 2.
B. Estimate the default risk premium.
C. Calculate upper and lower limits for the interest rate on Investment 3, r3.
2. A client has a $5 million portfolio and invests 5 percent of it in a money market fund projected to earn 3 percent annually. Estimate the value of this portion of his portfolio after seven years.
3. A client invests $500,000 in a bond fund projected to earn 7 percent annually. Estimate the value of her investment after 10 years.
4. For liquidity purposes, a client keeps $100,000 in a bank account. The bank quotes a stated annual interest rate of 7 percent. The bank's service representative explains that the stated rate is the rate one would earn if one were to cash out rather than invest the interest payments. How much will your client have in his account at the end of one year, assuming no additions or withdrawals, using the following types of compounding?
A. Quarterly
B. Monthly
C. Continuous
5. A bank quotes a rate of 5.89 percent with an effective annual rate of 6.05 percent. Does the bank use annual, quarterly, or monthly compounding?
6. A bank pays a stated annual interest rate of 8 percent. What is the effective annual rate using the following types of compounding?
A. Quarterly
B. Monthly
C. Continuous
7. A couple plans to set aside $20,000 per year in a conservative portfolio projected to earn 7 percent a year. If they make their first savings contribution one year from now, how much will they have at the end of 20 years?
8. Two years from now, a client will receive the first of three annual payments of $20,000 from a small business project. If she can earn 9 percent annually on her investments and plans to retire in six years, how much will the three business project payments be worth at the time of her retirement?
9. To cover the first year's total college tuition payments for his two children, a father will make a $75,000 payment five years from now. How much will he need to invest . today to meet his first tuition goal if the investment earns 6 percent annually?
10. A client has agreed to invest €100,000 one year from now in a business planning to expand, and she has decided to set aside the funds today in a bank account that pays
7 percent compounded quarterly. How much does she need to set aside?
11. A client can choose between receiving 10 annual $100,000 retirement payments, starting one year from today, or receiving a lump sum today. Knowing that he can invest at a rate of 5 percent annually, he has decided to take the lump sum. What lump sum today will be equivalent to the future annual payments?
12. A perpetual preferred stock position pays quarterly dividends of $1,000 indefinitely (forever). If an investor has a required rate of return of 12 percent per year on this type of investment, how much should he be willing to pay for this dividend stream?
13. At retirement, a client has two payment options: a 20-year annuity at €50,000 per year starting after one year or a lump sum of €500,000 today. If the client's required rate of return on retirement fund investments is 6 percent per year, which plan has the higher present value and by how much?
14. You are considering investing in two different instruments. The first instrument will pay nothing for three years, but then it will pay $20,000 per year for four years. The second instrument will pay $20,000 for three years and $30,000 in the fourth year. All payments are made at year-end. If your required rate of return on these investments is
8 percent annually, what should you be willing to pay for:
A. The first instrument
B. The second instrument (use the formula for a four-year annuity)
15. Suppose you plan to send your daughter to college in three years. You expect her to earn two-thirds of her tuition payment in scholarship money, so you estimate that your payments will be $10,000 a year for four years. To estimate whether you have set aside enough money, you ignore possible inflation in tuition payments and assume that you can earn 8 percent annually on your investments. How much should you set aside now to cover these payments?
16. A client is confused about two terms on some certificate-of-deposit rates quoted at his bank in the United States. You explain that the stated annual interest rate is an annual rate that does not take into account compounding within a year. The rate his bank calls APY (annual percentage yield) is the effective annual rate taking into account compounding. The bank's customer service representative mentioned monthly compounding, with $1,000 becoming $1,061.68 at the end of a year. To prepare to explain the terms to your client, calculate the stated annual interest rate that the bank must be quoting.
17. A client seeking liquidity sets aside €35,000 in a bank account today. The account pays 5 percent compounded monthly. Because the client is concerned about the fact that deposit insurance covers the account for only up to €100,000, calculate how many months it will take to reach that amount.
18. A client plans to send a child to college for four years starting 18 years from now. Having set aside money for tuition, she decides to plan for room and board also. She estimates these costs at $20,000 per year, payable at the beginning of each year, by the time her child goes to college. If she starts next year and makes 17 payments into a savings account paying 5 percent annually, what annual payments must she make?
19. A couple plans to pay their child's college tuition for 4 years starting 18 years from now. The current annual cost of college is C$7,000, and they expect this cost to rise at an annual rate of 5 percent. In their planning, they assume that they can earn 6 percent annually. How much must they put aside each year, starting next year, if they plan to make 17 equal payments?
20. You are analyzing the last five years of earnings per share data for a company. The figures are $4.00, $4.50, $5.00, $6.00, and $7.00. At what compound annual rate did EPS grow during these years?
SOLUTIONS 1. A. Investment 2 is identical to Investment 1 except that Investment 2 has low liquid ity. The difference between the interest rate on Investment 2 and Investment 1 is 0.5 percentage point. This amount represents the liquidity premium, which represents compensation for the risk of loss relative to an investment's fair value if the investment needs to be converted to cash quickly.
B. To estimate the default risk premium, find the two investments that have the same maturity but different levels of default risk. Both Investments 4 and 5 have a maturity of eight years. Investment 5, however, has low liquidity and thus bears a liquidity premium. The difference between the interest rates of Investments 5 and 4 is 2.5 percentage points. The liquidity premium is 0.5 percentage point (from Part A). This leaves 2.5 - 0.5 = 2.0 percentage points that must represent a default risk premium reflecting Investment 5's high default risk.
C. Investment 3 has liquidity risk and default risk comparable to Investment 2, but with its longer time to maturity, Investment 3 should have a higher maturity premium. The interest rate on Investment 3, r3, should thus be above 2.5 percent (the interest rate on Investment 2). If the liquidity of Investment 3 were high, Investment 3 would match Investment 4 except for Investment 3's shorter maturity. We would then conclude that Investment 3's interest rate should be less than the interest rate on Investment 4, which is 4 percent. In contrast to Investment 4, however, Investment 3 has low liquidity. It is possible that the interest rate on Investment 3 exceeds that of Investment 4 despite 3's shorter maturity, depending on the relative size of the liquidity and maturity premiums. However, we expect r3 to be less than 4.5 percent, the expected interest rate on Investment 4 if it had low liquidity. Thus 2.5 percent < r3 < 4.5 percent.
$250,000 X
ii. Identify the problem as the future value of a lump sum.
iii. Use the formula for the future value of a lump sum. PV = 0.05 X $5,000,000 = $250,000
$250,000 $307,468.47
PV FV
The future value in seven years of $250,000 received today is $307,468.47 if the interest rate is 3 percent compounded annually.
$500,000
pv x
ii. Identify the problem as the future value of a lump sum.
iii. Use the formula for the future value of a lump sum. FVa, = PV(1 + rf
$983,575.68 FV
$500,000 PV
Your client will have $983,575.68 in 10 years if she invests $500,000 today and earns 7 percent annually.
4. A. To solve this problem, take the following steps:
i. Draw a time line and recognize that a year consists of four quarterly periods.
$100,000 x
PV FV
ii. Recognize the problem as the future value of a lump sum with quarterly compounding.
iii. Use the formula for the future value of a lump sum with periodic compounding, where m is the frequency of compounding within a year and N is the number of years.
0.07
$100,000 PV
$107,185.90 FV
iv. As an alternative to Step iii, use a financial calculator. Most of the equations in this chapter can be solved using a financial calculator. Calculators vary in the exact keystrokes required (see your calculator's manual for the appropriate keystrokes), but the table below illustrates the basic variables and algorithms.
Time Value of Money Variable
Notation Used on Most Calculators
Numerical Value for This Problem
Number of periods or payments Interest rate per period Present value Future value Payment size
FV compute
$100,000
Remember, however, that a financial calculator is only a shortcut way of performing the mechanics and is not a substitute for setting up the problem or knowing which equation is appropriate.
In summary, your client will have $107,185.90 in one year if he deposits $100,000 today in a bank account paying a stated interest rate of 7 percent compounded quarterly.
B. To solve this problem, take the following steps:
i. Draw a time line and recognize that with monthly compounding, we need to express all values in monthly terms. Therefore, we have 12 periods.
$100,000 PV
ii. Recognize the problem as the future value of a lump sum with monthly compounding.
iii. Use the formula for the future value of a lump sum with periodic compounding, where m is the frequency of compounding within a year and N is the number of years.
$107,229.01
$100,000 PV
$107,229.01 FV
iv. As an alternative to Step iii, use a financial calculator.
Notation Used on Numerical Value
Most Calculators for This Problem
N 12
PV $100,000
FV compute X
Using your calculator's financial functions, verify that the future value, X, equals $107,229.01.
In summary, your client will have $107,229.01 at the end of one year if he deposits $100,000 today in his bank account paying a stated interest rate of 7 percent compounded monthly.
C. To solve this problem, take the following steps:
i. Draw a time line and recognize that with continuous compounding, we need to use the formula for the future value with continuous compounding.
$100,000 x
PV FV
ii. Use the formula for the future value with continuous compounding (N is the number of years in the expression). FVjv = PV J'N
The notation ea07(1) is the exponential function, where e is a number approximately equal to 2.718282. On most calculators, this function is on the key marked e*. First calculate the value of X. In this problem, X is 0.07(1) = 0.07. Key 0.07 into the calculator. Next press the ex key. You should get 1.072508. If you cannot get this figure, check your calculator's manual.
$100,000 $107,250.82
PV FV
In summary, your client will have $107,250.82 at the end of one year if he deposits $100,000 today in his bank account paying a stated interest rate of 7 percent compounded continuously.
5. Stated annual interest rate = 5.89 percent.
Effective annual rate on bank deposits = 6.05 percent.
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